A Virtual Finite Element Method for Two Dimensional Maxwell Interface Problems with a Background Unfitted Mesh

Abstract

A virtual element method (VEM) with the first order optimal convergence order is developed for solving two dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual H(curl)-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.

Immersed Virtual Element Methods for Elliptic Interface Problems

“…Remark 3.5. The global Nédélec edge element constructed in the proof above can be understood as a function in the virtual space developed in [13] with discontinuous coefficients, in which the DoFs associated with the internal edges of an interface element are eliminated by imposing a single constant curl value.…”

“…Lemma 4.5 (Lemma 4.2 in [28] and Lemma 5.4 in [13]). For u ∈ H 1 (curl, α, β; T h ), on any K ∈ T i h , the difference of the extensions on the approximate interface Γ K h along the tangential direction t satisfies (4.11)…”

“…. Following [13], using the same β e in (5.3), we directly employ the DoFs to construct the stabilization S e K (•, •) :…”

“…Remark 6.1. Note that the scaling in (6.2) is different from the conventional VEM using h 1/2 [13,8,22]. On the non-interface triangles, because the global space in (3.12) uses conforming Nédélec elements, the stabilization term in (6.1) vanishes.…”

“…As the interface may intersect elements arbitrarily which generates elements with high aspect ratio, for the aforementioned approach in [16], one major difficulty is to obtain a robust a priori error estimate independent of the potential anisotropic subelement shapes. Some anisotropic error analysis of VEM can be found in [11,12,13] for different interface problems.…”